For school I need to set up an HTML5 live stream site. They have a flash stream-player they've been using but now they want it to use HTML5 instead. How can I do this? I tried using the video tag but I can't get it working. Below is the code I have. Could someone point me in the correct direction?

Live streaming in HTML5 is possible via the use of Media Source Extensions (MSE) - the relatively new W3C standard: https://www.w3.org/TR/media-source/
MSE is an an extension of HTML5 <video> tag; the javascript on webpage can fetch audio/video segments from the server and push them to MSE for playback. The fetching mechanism can be done via HTTP requests (MPEG-DASH) or via WebSockets. As of September 2016 all major browsers on all devices support MSE. iOS is the only exception.

For high latency (5+ seconds) HTML5 live video streaming you can consider MPEG-DASH implementations by video.js or Wowza streaming engine.

For low latency, near real-time HTML5 live video streaming, take a look at EvoStream media server, Unreal media server, and WebRTC.

It is not possible to use the RTMP protocol in HTML 5, because the RTMP protocol is only used between the server and the flash player. So you can use the other streaming protocols for viewing the streaming videos in HTML 5.

Firstly you need to setup a media streaming server. You can use Wowza, red5 or nginx-rtmp-module. Read their documentation and setup on OS you want.
All the engine are support HLS (Http Live Stream protocol that was developed by Apple). You should read documentation for config.
Example with nginx-rtmp-module:

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